ICOSAHEDRAL SYMMETRY and the QUINTIC EQUATION Typeset by A

Home , Algebraic solution, Sextic equation

ICOSAHEDRAL SYMMETRY AND THE QUINTIC EQUATION

R. B. KIN(] Department of Chemistry, Uuivendty of Georgia Athe~m, Georgia 30602, U.S.A. E. R. CANFIELD Department of Computer Science, University of Georgia Athens, Georgia 30602, U.S.A.

(Received J~lll 1990)

A.bstract--An algorithm has been implemented on a microcomputer for solving the general quintic equation. This algorithm is bam~cl on the isomorphism of the As alternating Galois group of the gcmeral quintic equation to the symmetry group of the Jcosahedron, coupled with the ability to partition an object of ieosahedral symmetry into five equivalent objects of tetrahedral or octabedral symmetry. A detailed discussion is presented of the portions of this algorithm relating to these symmetry properties of the icosahedron. In addition, the entire algorithm is summarized.

I. INTRODUCTION One of the classicalproblems of algebra is the solution of polynomial equations using an algorithm which calculates the roots of the equation from its coefficients.Such a formula for the roots of a quadratic equation is well-known to even beginning algebra students and contains a single square root. An analogous but considerably more complicated algorithm containing both square and cube roots is known for solution of the cubic equation [1,2]. Appropriate combination of the algorithms for the quadratic and cubic equations leads to an algorithm for solution of the quartic equation [1,2]. However, persistentefforts by numerous mathematicians before the early 1800's to find an algortithm for solution of the general quintic equation using only radicals and elementary arithmetic operations were uniformly unsuccessful. The reason for the futilityof these efforts was uncovered by Galois [3-5], who proved the impossibilityof solving a general quintic equation using only radicals and arithmetic operations. Galois' proof led to the birth of group theory and the concept of the symmetry of an algebraic equation relating to permutations of its roots [3-5]. The demonstrated impossibility of solving quintic equations using only radicals and simple arithmetic led the 19th century mathematicians following Galois to develop the mathematics of the more complicated functions required for an algebraic solution of the quintic equation. The necessary functions turn out to be Jacobi and/or Weierstrass ellipticfunctions [6] and the theta series used for their evaluation. The required relationships between such ellipticfunctions and the roots of a class of quintic equations derivable from the general quintic equation by Tschirn- haus transformations [7] were first established by Hermite [8,9], and developed subsequently by Gordan [10] into an algorithm for solution of the quintic equation. This algorithm was both simplified and clarified by Kiepert [11]. The nature of these algebraic efforts was clarified further by publication, in 1884, of Klein's classical book linking the symmetry of the icosahedron with algorithms for solution of the quintic equation [12]. This book attracted the attention of one of us (R.B.K), who for many years has been fascinated by the special features of icosahedral symmetry [13] in chemical contexts, such as chirality [14], boron hydrides [15] and quasicrystals [16]. These 19th-century algorithms for the solution of the general quintic equation were so complicated that there was no hope of implementing them in that precomputer era. Numerical,

We are indebted to the U.S. Office of Naval Research for partial support of this work.

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13 14 R.B. KINa, E.R. CANFIELD rather than algebraic methods based on elliptic functions, evolved to satisfy practical needs for determining roots of quintic and higher degree equations [17]. However, such methods ignore the interesting symmetry aspects of such equations first discovered by Galois [3-5]. The concise presentation of an apparent algorithm for solution of a general quintic equation at the end of the 1878 paper by Kiepert [11] stimulated us to try this algorithm on a modern microcomputer. However, our initial attempts soon revealed a number of ambiguities and even errors of the ex- pected type for an algorithm which had never been actually implemented because of the lack of the necessary computer technology at the time of its development. We therefore soon real- ized that, contrary to the impression given by Cole [18] as long ago as 1886, a really complete workable algorithm for solution of the general quintic equation by elliptic functions, rather than numerical methods, was not contained in any of the 19th century or more recent literature. We therefore undertook a deeper study of the underlying rather complicated mathematics. Useful surveys of much but not all of the necessary algebra for solution of the general quintic equation are presented in 20th century, but now rather old, algebra texts by Dickson [19] and Perron [20]. However, even these less ancient writeups lack much of the necessary information for a complete algorithm for solution of the general quintic equation. We have now filled in all of the gaps and have a complete computer-tested algorithm for solution of the general quintic equation by algebraic rather than numerical methods. The underlying mathematics of this successful algorithm is long and complicated. This paper focusses on the aspects of the algorithm based on the icosahedral symmetry of the general quintic equation. In addition, a summary of the complete algorithm is presented, so that it can be implemented by an interested reader on an appropriate microcomputer without referring to the literature. Future publications are planned to discuss other aspects of this algorithm.

2. THE SYMMETRY OF ALGEBRAIC EQUATIONS Consider an algebraic equation of degree n

f(x) = a0x n + al x "-t +"'+ an = Za"-iz i = 0, (2.1) i----1 where ak (0 < k <: n) is rational. Without loss of generality, a0 can be taken to be unity since ak (1 <__ k _< n) can be adjusted appropriately. The polynomial f(z) is said to be factorable if it can be expressed as a product of polynomials of lower degrees with rational coefficients. The Galois group, G, of an algebraic equation is a permutation group of its roots [3-5]. If the polynomial of the equation is non-factorable, then the corresponding Galois group, G, is transitive, i.e., for any pair of roots zi and zk, there is an operation g in G such that zi is permuted to xk. The structure of the Gaiois group, G, of an algebraic equation relates to the types of functions that are needed to determine its roots. Consider a group G and a subgroup H, with g E G and h E H. If g is fixed and h is varied over the elements of H, then all elements of the form g hg -1 form another subgroup of G which is said to be conjugate to H. A normal subgroup N with n E N is one which is self-conjugate, i.e., g ng -1 generates N, and we write N~G. Subgroups H induce an equivalence relation on the group G partitioning it into various equivalence classes, two elements g and g~ of G being regarded as equivalent when g (g~)-t E H. A normal subgroup N as defined above consists of entire conjugacy classes of G. The conjugacy class headings of the character table of G are thus useful for finding the normal subgroups of G. If N is a normal subgroup of G, the quotient group G/N is well-defined and relates to the type of functions needed to determine the roots of an equation with the Galois group G. A group G with only the identity group C1 as a normal subgroup is called a simple group. Simple permutation groups tend to be groups of high order relative to the number of objects being permuted. For example, the alternating groups An, consisting of all n!/2 even permutations of n objects, are simple groups for n >_ 5. The smallest non-trivial simple permutation group is thus the alternating group As, which is isomorphic with the icosahedral pure rotation group / [13]. This is the general basis of the relationship of icosahedral symmetry to the solution of the general quintic equation, as Icoaahedr~l symmetry 15 summarized in Klein's book [12]. The icosahedral pure rotation group I is the only non-trivial simple group which can serve as a symmetry point group in three dimensions. This accounts for the unusual appearance of its character table [21]. Consider a group G that is not simple. Then a series of a maximum number of normal subgroups can be assembled, so that C1 = Go ,~ G1 "~"" "~ G,n = G, (2.2) and each factor group Gi/Gi_l is simple. The group G is said to be solvable when each of these factors is of prime order. In this case, each step from Gi to Gi-1 in (2.2) generates radicals of the type ~/~ in the solution of an equation with G as the Galois group, where s is the number of elements in the factor group Gi/Gi-1. Thus equations with solvable Galois groups are exactly those which are solvable using radicals. Conversely, general equations of degree 5 or higher, which have the simple alternating groups An (n > 5) as their Galois groups, are not solvable using only radicals. Some of these and other relevant ideas can be illustrated by the well-known algorithm for solution of the general cubic equation [2]

z a-l-a1 x 2 ÷ a2z+a3 = 0. (2.3)

The Galois group of Equation (2.3), Sa, is isomorphic to the Dab point group of the equilateral triangle, and has the following normal series and associated factor groups:

Go = C1 ~ Ca = Aa '~ Sa = Daa, (2.4)

A_s = c3, (2.5a) CI $3 _ C2. (2.5b) Aa

The solution of a general cubic equation based on this normal series is thus seen to involve a square root from the factor group Ss/A3 = C2, inside the cube root from the factor group As/C1 = Ca. In the actual algorithm to solve such a general cubic equation (2.3), the equation is first simplified by applying a Tschirnhaus transformation [7] to give the following "reduced" cubic equation with no quadratic term: z 3 -t- b2 z -t- b3 -- O, (2.6) where

X~Z-- ~a1~ (2.7a) 1 b2 = ] (3 2 - (2.7b)

bs = ~ (2al8 - 9ai a~ + 27 as). (2.7c)

The roots, zk (k = 0, 1, 2), of the "reduced" cubic equation (2.6) can be expressed by the relations

(2.8)

with the cube roots of the two terms being selected so that their product is always -b~/3. If the discriminant 4 b] + 2? b~ is negative, then the corresponding cubic equations (2.8), and hence (2.3), have three distinct real roots, but the evaluation of these real roots requires taking cube 16 R.B. KING, E.R. CANFIELD roots of imaginary numbers [i]. In order to avoid this difficulty,the following trigonometric formulas are used to determine the roots of a cubic equation with a negative discriminant [1]:

¢~ --" COS--1 (2.9a) ( 2 (2.9b)

Note that the sequence of steps (2.9a) and (2.9b) first involves taking the inverse cosine of one number followed by taking the cosine of another number, derived from the first inverse cosine. Both the logarithms needed in general to evaluate the radicals for Equation (2.8) and the inverse cosine needed for Equation (2.9a) can be expressed as integrals of the type ,,(a) = jr1a dy (2.10) where g(y) is a quadratic polynomial, i.e., y2 for loga, and 1 -y2 for -cos -1 a. Integrals of the type (2.10), but with cubic or still higher degree polynomials for g(y), are used for solution of general equations of degree 5 or higher having simple Galois groups. Thus the solvability of an equation by radicals can be translated into the solvability using integrals of the type (2.10) with a quadratic polynomial g(y) in the denominator, whereas the inability to solve an equation using only radicals means that the polynomial g(y) in the denominator of integral (2.10) must be of degree 3 or higher. Inverses of integrals of the type (2.10) in which g(y) is cubic or quartic correspond to the Weierstrass or Jacobi elliptic functions [22], respectively, which play key roles in the solution of the general quintic equation discussed in this paper. Increasing the degree of an algebraic equation from 3 to 4 introduces no fundamentally new features. The Galois group of the general quartic, $4, is isomorphic to the Td full point group of the regular tetrabedron and has the following normal series and associated factor groups [2]:

Go -" C1 ,~ C2 ,~ D2 ,~ A4 = T ,~ S4 = Td, (2.11)

C'-K2 - C2, (2.12a) CI D.._.~2_- C~, (2.12b) C2 A...~4= C3, (2.12c) D2 S--!4= C2. (2.12d) A4 Thus, the solution of a quartic equation can be reduced to the solution of a resolvent cubic equation (using (2.8) or (2.9)) combined with two additional quadratic equations in accord with the standard algorithms [1,2]. The general quintic equation with the $5 Galois group exhibits important fundamentally new features because of the simplicity of the A5 group in its normal series and associated factor groups, namely, Go = C1 ,~A5 = I ,~Ss, (2.13)

A_..~5 -- As, (2.14a) C1 S__~.5= c2. (2.14b) A5 Icosahedr&l symmetry 17

B 0

C Figure 1. Partitioning the 20 vertices of a regu- Figure 2. A Schlegel diagram of a regular lar dodecahedron into five sets of four vertices, dodecahedron, showing the relative positions each correspovd|n~ to regular tetrahedra. The of all five sets of tetrahedral vertices. four vertices in one of these sets are circled.

Figure 3. The icosidodecahedron formed from the 30 edge-mldpoints of a regular icosahedron, an the partitioning of its 30 vertices into five sets of six vertlces, each corresponding to regular octahedra. The six vertices in one of these sets are circled.

The simplicity of the Au factor group, which is isomorphic to the icosahedral pure rotation group, I, precludes purely radical solutions of the general quintic equation as noted above. How- ever, the ability to partition an appropriate object of icosahedral symmetry into five equivalent objects of tetrahedral or octahedral symmetry provides the key to the algorithm for solution of the general quintic equation discussed in this paper. There are several ways of visualizing this type of partition of an object of icosahedral symmetry into five equivalent objects of at least tetrahedral symmetry, thereby separating the effect of the five-fold axis of the icosahedron from that of its two-fold and three-fold symmetry elements. The 20 vertices of a regular dodecahedron can be partioned into five sets of four vertices, each corresponding to a regular tetrahedron. One of these tetrahedral sets of four vertices is circled on the dodecahedron in Figure 1 and all five sets of tetrahedral vertices are labeled in the Schlegel diagram of a regular dodecahedron in Figure 2. Klein [12] partitions the 30 edges of an icosahedron into five sets of six edges each by the following method: (1) A straight line is drawn from the midpoint of each edge through the center of the icoss- hedron to the midpoint of the opposite edge; (2) The resulting 15 straight lines are divided into five sets of three mutually perpendicular straight lines. 18 R.B. KINO, E.R. CANFIELD

Each set of three mutually perpendicular straight lines resembles a set of Cartesian coordinates and defines a regular octahedron. This construction is the dual [23] of a construction depicted by DuVal [24] in color, in which a regular dodecahedron is partioned into five equivalent cubes. Alternatively, the 30 edge midpoints of an icosahedron can be used to form an icosidodecahedron (Figure 3) having 30 vertices, 60 edges, 32 faces, and retaining icosahedral symmetry. The 30 vertices of this icosidodecahedron can then be partitioned into five sets of six vertices, each corresponding to a regular octahedron. One of these sets of octahedral vertices is circled on the icosidodecahedron in Figure 3. The approach of Kiepert [11] for solving the general monic quintic equation

zS + Az4 + Bz3 +Cz2 + Dz+ E=O, (2.15) which is the basis of our complete computer-tested algorithm, can be broken down into the following seven steps: (1) Tschirnhaus transformation of the general quintic, into the principal quintic

zS W 5az 2 + 5bz Wc = O. (2.16)

This Tschimhaus transformation requires only a single square root in addition to arithmetic operations. (2) A second Tschirnhans transformation of the principal quintic (2.16) into the Brioschi quintic y5 _ 10 Z y3 + 45 Z 2 y - Z 2 = 0, (2.17) in which the coefficients are expressed in terms of a single parameter Z. This Tschirnhaus transformation also requires only a single square root in addition to simple arithmetic operations. This is the step of the Kiepert algorithm which uses the special symmetry properties of the icosahedron, including the ability to partition an object of icosahedral symmetry into five equivalent objects of at least tetrahedral symmetry, as noted above. The next section of this paper discusses this step in detail, since this is the critical step relating icosahedral symmetry to the quintic equation as already noted by Klein [12] in his 1884 book on the icosahedron. (3) Transformation of the Brioschi quintic (2.17) into the Jacobi sextic

I0 s 12 g2 5 s ~+~-s - Am-F -s+~=0, (2.18) 1 where Z - ---. (2.19) A The roots soo and sk (0 < k < 4) of the Jacobi sextic can be used to calculate the roots, Yk (0 < k <4), of the Brioschi quintic (2.17), by the relationship

1 ] 1/~ , (2.2o)

with addition of indices modulo 5. The Galois group of the Jacobi sextic (2.18) has 60 elements, like the A5 Galois group of the quintic equations (2.15), (2.18) and (2.17), but involves transitive permutations of the six roots sco and sk (0 ~ k _< 4) rather than only the five roots of the quintic equations [20]. (4) Solution of the Jacobi sextic (2.18) by the Weierstrass elliptic functions defined by the integral (2.10) with a cubic polynomial in the denominator, defined by

= 4p - g2 p - g3, (2.21)

where g2 is the same as the g2 in (2.18), and g3 is related to g2 and A in (2.18). Icosahedr&l symmetry 19

(5) Evaluation of these Weierstrass elliptic functions using rapidly converging genus 1 theta functions [6,25]. (6) Evaluation of the periods of the theta functions corresponding to a particular Jacobi sextic (2.18). This requires solution of the cubic equation g(y) = O, (2.22)

where g(y) is defined by Equation (2.21) with the parameters g2 and gs coming from the g2 and A in the Jacobi sextic (2.18). A simple function of the three roots of the cubic (Equations (2.21) and (2.22)) is then substituted into a special infinite series called the Jacobi home [26]. Substitution of these periods into appropriate theta series gives the roots soo and s~ (0 < k < 4) of the Jacobi sextic (2.18). (7) Calculation of the five roots, Yk, of the Brioschi quintic (2.17), from the roots soo and sk (0 < k < 4) of the Jacobi sextic (2.18), using (2.20) and then undoing the Tschirnhaus transformations of the quintic equations in the sequence Brioschi (2.17) , Principal (2.16) , General (2.15). (2.23)

The next section presents details of the second step of this algorithm, which pertains to the icosahedral symmetry of the general quintic equation (2.15). The final section of this paper summarizes the entire algorithm in sufficient detail for programming on a computer. 3. THE ROLE OF ICOSAHEDRAL SETS OF OCTAHEDRA IN THE TSCHIRNHAUS TRANSFORMATION OF THE PRINCIPAL QUINTIC TO THE BRIOSCHI QUINTIC The objective of this step of the algorithm for the solution of the general quintic equation (2.15) is a Tschirnhaus transformation of a principal quintic (Equation (2.16)) z 5+5az 2+5bz+c= O, into a single parameter Brioschi quintic (Equation (2.17))

yS_ 10ZyS+45Z2y_ Z ~ = O. This transformation uses the following relationship between the variables in (2.16) and (2.17):

A + p y (3.1) = C 2lZ) - 3" This Tschirnhaus transformation is best described geometrically using polyhedral polynomials on the Riemann sphere [12,19]. The five roots of the Brioschi quintic (2.17) are thus the vertex polynomials of the five octahedra in an icosahedral set, where the single parameter Z in Equation (2.17) is obtained from the polyhedral polynomials of the underlying icosahedron. The genesis and properties of the relevant polyhedral polynomials are now presented in greater detail. Consider a stereographic projection of the north pole of a unit sphere p2 + q~ + r~ = 1 (the Riemann sphere) onto its equatorial plane as an Argand plane. A complex number z = a + bi gives a'- P q a + bi = p + iq (3.2) 1-r' b= 1-r' -'-~"1 Solving for p, q and r gives 2a 2b -1 + a 2 + b2 r = (3.3) P= l+a2+b~, q= l +a2 +b 2, l +a2 +b ~ • The north pole of the sphere corresponds to c¢ and the south pole to 0. Every rotation of this sphere around its center corresponds to a linear substitution

a z + ~ (3.4) z' = 7z +'-----~" 20 R.B. Kmo, E.R. CANFIICLD

For a rotation of the sphere by an angle #, where p, q, r and -p, -q, -r remain constant,

= (,, + • - (t - is) (3.5) (t + is) • + (v - iu)' where s = psin(O/2), t = q sin(0/2), u = r sin(O~2) and v = cos(O/2), so that

s 2 + t 2 + u 2 + v 2 = 1. (3.6)

For a rotation about the O-co axis (south pole-north pole), this reduces to

=' = ei'z. (3.7)

Consider a regular octahedron and a regular icosahedron represented by points on the surface of such a Pdemann sphere so that the north pole (z = co) is one of the vertices in each case. This leads to the following homogeneous polynomials for the regular octahedrou and icosahedron in these orientations where z is taken to be u/v [12,19]:

(a) Octahedron (04 symmetry): Vertices: r - u ~ (u 4 - v4), (3.8a) Edges: X = u12 - 33 u s v 4 - 33 u 4 v s + v 1~, (3.8b) Faces: W = u s -t- 14 u 4 v 4 + v s. (3.8c) (b) Icosahedron (I4 symmetry): Vertices: f = u v (u l° + 11 u ~ v 5 - rio), (3.9a) Edges: T = u 3° + 522 u 25 v 25 - 10,005 u 2° v 1° - 10,005 u 1° v20 _ 522 u 5 v 25 + ~8o, (3.9b)

Faces: H = -u 2° + 228 u 15 v 5 - 494 u I° v 1° - 228 u 5 v 15 - v 2°. (3.9c)

The polynomials in Equations (3.8) and (3.9) are conveniently called polyhedral polynomials. Their roots correspond to the locations of the vertices, edge midpoints and fsce midpoints on the PAemann sphere. Their degrees are equal to the numbers of corresponding elements (vertices, edges, or faces). The special symmetries of the regular octahedron and icosahedron lea~! to the following identities:

Octahedron: 108 ~.4 _ W3 + X2 = 0, (degree 24), (3.10a) Icosahedron: 1728 fs _ H a _ T 2 = 0, (degree 80). (3.10b)

Equation (3.10b) plays a role in the Tschirnhaus transformation of a principal quintic (2.16) into the corresponding Brioschi quintic (2.17). Note that the degrees of the left sides of the identities (3.10a) and (3.10b) in (u, v) (see (3.8) and (3.9)) correspond to the orders of the pure rotation groups of the corresponding polyhedra which are isomorphic to permutation groups relevant to the solution of algebraic equations, i.e., O ~ $4 for the quartic equation, and I ~ As for the quintic equation, respectively. The special symmetries of these regular polyhedrs also ]end to the vanishing of the fourth transvectants of the vertex po]ynonda]s (I-, ~.)4 and (f, j,)4 as determined by the methods of invariant theory [27,28]. Consider, further, the icosahedron on a Riemann sphere so that one vertex is at the north pole. The midpoints of its 30 edges define an icosahedral set of five octshedra with the following vertex polynondals, t~, and face polynomials, Wk, (0 < k _< 4) [19]:

tk = ~3kue -t- 2e2kuSv - 5eku4~ 2 - 5e 4tu2v4 - 2 (atuvs -6 e:kv6, (3.11a) Wk = -e~u s + e3ku7v - 7e=ku% = - 7eku% 3 + 7 e4*u%s - 7eSku%e - (2kuv7 -- ekvs" (3.11b) Icou~edralsymmetry 21

In Equations (3.11), e = exp(2~ri/5) and 0 _< k _< 4. Furthermore, the octahedral polynomi- Ms (3.11) are different from the octahedral polynomials (3.8) because of the different orientations of the octahedra relative to the 0-oo axis. The vertex polynomials tk (Equation (3.11a)) of an icosaheclral set of five oetahedra play an important role in the Tschirnhans transformation of a principal quintic (2.16) into the Brioschi quintie (2.17), by being the roots of a closely related Briosehi quintic

t 5 - 10ft 3 + 45f2t - T = 0, (3.12) in which f and T are the vertex and edge functions of the underlying icosahedron defined by (3.9a) and (3.9b), respectively. Substituting f2 = T into (3.12) and taking t = yT/f ~ leads to the one- parameter Briosehi quintic (2.17). Now consider the principal quintic (2.16). Express its roots, zk, using the polyhedral polynomials (3.11) of the icosahedral set of five octahedra, in conjunction with the polyhedral polynomials (3.9) of the underlying icosahedron, to give

zk = Wk + kH'T/t~ Wk, (0 < k < 4). (3.13)

In (3.13), f, H and T are the icosahedral polynomials (3.9), and t~ and Wk are the polynomials of the ieosahedral set of five oetahedra (Equations (3.11)). The right hand side of Equation (3.13) is thus homogeneous of degree zero in the variable pair (u, v). The ability to express the roots of any principal quintie (2.16) by (3.13) depends upon the following properties of the polyhedral polynomials, which lead to the vanishing of the z 4 and z s eoemeients in the principal quintie (2.16):

E Wk = O, (degree 8), (3.14a) EtkWk = 0, (degree 14), (3.14b) E W~ = 0, (degree 16), (3.14c) Et~W~ = O, (degree 22), (3.14d) tkW~ 2 = O, (degree 28). (3.14e)

The vanishing of the sums in Equations (3.14) is a consequence of the fact that no product of the icosahedral polynomials f, H and T, of degrees 12, 20 and 30, respectively, can have degrees 8, 14, 16, 22 and 28. The next step is the calculation of the coefficients a, b and c of the principal quintie (2.16) with roots zk expressed by (3.13), in terms of two composite icosahedral functions Z and V where

f5 Z = T---~, (3.15a) H s V = f--g-, (3.15b) and Z and V are related by 1 + V = 1728, (3.16) derived from the identity (3.10b). This calculation uses the following powers of the oetahedral polynomials tk and Wk of which the first five terms are given:

t~ = ek u 12 + 4u 11 v -- 6e4k u 10 v 2 - 20 eak u 9 v a + 15 e2k u s v 4 + ... , (3.17a) t3 __ £4k ulS ..1_ 6~3k U17 I) -- 3 e 2k u 16 v ~ - 52 ek U 15 V3 -I- 84 e4k 1113 I~5 dr .'. , (3.17b) no u14~) 4 term) t'~ = e ~k u 24 + 8 ek u 2s v + 4 u ~ v ~ - 88 ~4k u2t v 3 _ 94 esk u ~° v 4 +... , (3.17c) 22 R.B. KING, E.R. CANFIELD

W~ = eak u 16 - 2 e 2k u z5 v + 15 e k u 14 v 2 + 35e 4k u 12 v 4 + 84e sk U 11 t) 5 J¢" • • " , (3.18a) W~ = -e 2k U 24 "b 3 6 k u 2s v -- 24 u 22 v 2 + 22 e4k u 21 v s -- 126 esk u 2° v 4 +-", (3.18b) W 4 - ek u s2-4u m v +34e 4ku s°v 2- 60e a~ u 29v a+295e 2ku 2sv 4 +..- . (3.150

Also, note that if n is an integer not divisible by 5,

~.k = O, (3.19) k=l so that when taking sums of the type 5 t~ W~ =, (3.20) k=l all of the terms with the ejk (1 _< j < 4) coefficients in (3.18) and (3.19) vanish. By compar- ing the indicated terms with the indicated products of the icosahedral polynomials f, H and T (Equations (3.9)), the Equations (3.21) and (3.22) can be derived as follows:

u22v2 terms: EW~ -- (5)(--24) f2 _ -120 f~, (3,21a) u 30 terms: E W~tk = (5)(-i) T = -ST, (3.21b) uaSv3 terms: E W~ t~ = (5)(20 - 18 - 96 + 22) u~v s = (5)(-72) f3 _ -360 fS, (3.21c) U41V terms: E W~ t~ "-- (5) (--6 -{- 3) ?./41'/)--" (5) (--3) fT = -15 fT, (3.21d) terms: Z Wt = (5) (-4) u 31 v = (5) (4) f H - 20 f H, (3.22a) w:t, = 0, since no product of f (degree 12), H (degree 20) and, T (degree 30) can have degree 38, (3.22b) U42U 2 terms: ~ W~4tk2 = (5) (34 - 16 - 6) u42v2 = (5) (-12) f2H = -60 f2g, (3.22c) u 5° terms: E W~4 tk3 = (5) (-1) H T = -5 H T, (3.22d) USSVs terms: ~-~ W~4tk4 = (5) (-88 - 16 + 272 - 60) uSSvs = (5) (-108) fs H = -540 fs H. (3.22e)

In addition,

= -15 a, (3.23a) z =-20b (3 23h)

Calculating ~z~ and ~"~z~ by expanding (3.13) and substituting the values from (3.15) and (3.21)-(3.23) gives

V a = 8A s + A 2 ~ + (72 A ~2 + ~a) Z, (3.24) V b = -A 4 + 18 A 2 ~2 Z + A/~s Z + 27 p4 Z 2. (3.25) Icosahedral symmetry 23

Now consider Equation (3.2) and its five roots tk (0 _

x s - 10 f z a + 45 f s z - T = H(z - tk), (3.26) for all values of z. Let z = -~ Tip fs, which generates Equation (2.17) if y = -.~/p. Then,

H(AT+pf2tk) = ~5T5 - lO.~apSTaf~-F45.~p4Tfl° +psf1°T. (3.27)

However, H Wk -- -H s. (3.28) Then, after substituting the values for V and Z from (3.15), we get

Vc = h 5 - 10 h a ps Z + 45 ~ p4 Z 2 + p5 Z 2. (3.29)

Thus the values of a, b and c determined from (3.24), (3.25) and (3.29) give the coefficients of the principal quintic (2.16) corresponding to the Brioschi quintic (2.17), with the change of variables (3.1). Equations (3.24), (3.25) and (3.29), obtained above from the relationship of the polyhedral polynomials of the icosahedral set of five octahedra to those of the underlying icosahedron can now be inverted using a procedure outlined by Dickson [19], so that the parameters :~, p, Z and V can be calculated for the Brioschi quintic (2.17) corresponding to a given principal quintic (2.16) with coefficients a, b and c. First, add ~ Vb obtained from (3.25) to Vc from (3.29) to give /js Z V a as determined by Equation (3.24), i.e.,

p2Z a = )~ b + c. (3.30)

Then, substract p2 Z V b (obtained from (3.25)) from ~ V c (obtained from (3.29)), and use (3.30) to give V = (~2 _ 3ps Z)a (a ~2 _ 3b ~ - 3c) s (3.31) ()tc_ p2 Z b) - a2 (~ac_ )tbS _ bc ). Now, combine Equations (3.24) and (3.25) in the indicated manner to give

V (A a + 8 b) = ha + 216 A2 p Z + 9,~ ps Z + 216 pa Z s. (3.32) P Divide the square of the right hand side of (3.32) by Z, substract the result from 27 times the square of the right hand side of (3.24), and combine pairs of terms reflected by the identity (3.16) to give 27 a s V - V (;~ a + 8b) s p2 Z "- (A2 - 3ps Z)a" (3.33)

Now substitute (3.31) into (3.33), to give

27 a 2 (A a + 8b) 2 = ~ c - ps Z b. (3.34) p2 Z

Substitution of (A b + c)/a for pS Z from (3.30) now gives the following quadratic equation for A, in terms only of the coefficients a, b and c of the principal quintic (2.16):

AS(a4 +abe-b3)-A(llaab-acS + 2bSc)-b64a2bS- 27aSe-bes'-O. (3.35)

After solving this quadratic equation for ,~ the value for V can be found by (3.31), and then the value for Z from (3.16). Equation (3.24) can then be rewritten as

(A~ + ps Z) ~ = V a - 8A s - 72 A ~s Z. 24 I%.B. KING, E.I%. CANFIELD

Substitution of (3.30) for/z2Z into Equation (3.36), and solving for p then gives

Va 2 -- 8ASa -- 72A2b - 72Ac (3.37) P= )~2a+ )~b+ c In this way, the four parameters ~,p, V and Z are obtained for a Brioschi quintic (2.17) corresponding to a principal quintic (2.16) with roots (3.13) and coefficients a, b and c. The geometrical relationship between the five octahedra of an icosahedral set and the underlying icosahedron can be used to derive the Tschirnhaus transformation (3.1) relating the principal quintic (2.16) to the one-parameter Brioschi quintic (2.17). The octahedron face polynomials Wk, corresponding to the octahedron vertex functions tk which are solutions of the Brioschi quintic (3.12), vanish at the midpoints of the faces of the octahedra which are located at the midpoints of the faces of the underlying icosahedron, where its face polynomial H also vanishes. Hence, each Wk is a factor of H. In addition, transposing the term T of the quintic (3.12), squaring both sides, and replacing t ~- by 3f gives

1728 f5 = T 2. (3.38) Substituting the icosahedral identity (3.10b) into (3.38) gives H = 0 indicating that t~ - 3/is a factor of H for each k so that H = Wk (t~ - 3f), (0 < k < 4). (3.39)

Equation (3.39) thus defines five ways of factoring the degree 20 polynomial H into a degree 8 polynomial Wk and a degree 12 polynomial t~ - 3f. Substituting (3.39) into (3.13), suppressing subscripts, using (3.15a) for Z, and the relationship t = yT/f 2 to convert the two-parameter Brioschi quintic (3.12) into the one-parameter Brioschi quintic (2.17) gives the required relationship (3.1) between the variable z of the principal quintic (2.16) and the variable y of the Brioschi quintic (2.17).

4. THE ALGORITHM FOR SOLUTION OF THE GENERAL QUINTIC EQUATION Consider a general monic quintic (Equation (2.15))

z 5 +Az 4 + Bz s +Cz 2 + Dz + E = O. (4.1)

In order to apply the Tschirnhaus transformation (4.2) to give the corresponding principal quintic (Equation (2.16))

z s+5az 2 +Sbz+c= 0, (4.3) first obtain u by solving the quadratic equation

(2A 2 - 5B) u 2 + (4A8 - 13A B + 15C) u + (2A4 - 8A 2 B + 10A C + 3B 2 - 10 D) = 0. (4.4)

Then,

v = 1 (-A u - A 2 + 2B), (4.5)

a=I[-c(uS+Au2+Bu+C)+D(4u2+3Au+2B)-E(5u+2A)-IOrS], (4.6)

b = 5 [D (u4+A uS+S u2+C u+D) - E (5.S+4A .2+3B .+2C) - 5r4-10a v], (4.7) 1 C--" -E(u s+Au 4+B" 3+C" 2+Du+E)-v 5-5av 2-5by. (4.8) Icosahedral symmetry 25

Next apply the Tschirnhaus transformation (Equation (3.1)) A+py Z -- (4.9) (ys/Z) - 3' to the principal quintic (4.3) to give the one-parameter Brioschi quintic (Equation (2.17))

yS_ lOZy3 +45Z2y_ Z s = O, (4.10) by firstobtaining A, solving the quadratic equation (Equation (3.35))

(a 4 -{- a b c - b 3) A s - (11 a 3 b - a c 2 + 2b a c) A + 64 a s b s - 27 a s c - b c a = 0. (4.11)

Then (Equations (3.31), (3.16) and (3.37), respectively)

(a A 2 - 3b A - 3c)a (4.12) V-- aS(Aac_ Ab2_bc), 1 Z = (1728 - V)' (4.13) Va 2- 8A3a- 72A 2b- 72Ac (4.14) P= ASaWAb+c

Now calculate the elliptic invariants necessary to solve the corresponding 3acobi sextic (Equation (2.18)) sS g2 5 s 6+~-I0 - A--12 Y-s+~=0, (4.15) by the relationships

1 A = Z' (4.16)

g2 = ~/(1 - 1728Z)/Z 2 (4.17) 12

= I') (4.18) ¥ LI

In order to determine the periods of the theta functions needed to evaluate the relevant Weier- strass elliptic function, solve the cubic equation (Equation (2.21))

gCY) - 4Ps - gn P - gs = O, (4.19) by defining

1 G -- -~ g.q, (4.20a) 1 H = -~-~ gs. (4.20b) Then ],,3 U = (-G + ~/G ~ + 4H s) , (4.21)

H V = -~-, (4.22)

7 =exp (~). (4.23) 26 R.B. KING, E.R. CANFIELD

The three roots of g(y) (Equation (4.19)), designated as el, e2 and es, are obtained from U, V and 7 by

el = U + V, (4.24a) e2 = 7 U + 7 2 V, (4.24b) es = 72 U + '7 V. (4.24c)

Now assign the subscripts 1,2 and 3 to a,b and c, in such manner that neither

k2 = eb -- e¢, (4.25a) e a -- ee nor (k,) 2 = ea - eb , (4.25b) ea -- ee has a negative real part or a positive real part equal to or greater than unity. Now calculate

k' - V~a- eb eaVq:_L_.~_e¢ , (4.26) taking the square roots so that both k and k' have positive real parts. Next, calculate

L - (1 - Yr~) (4.27) (1 + x/'~)' taking the principal value of the square root V~'. The parameter q for the theta function can then be calculated by the Jacobi home [26] (L) (L) 5 (L) 9 (L) 13 (L) 17 (L) 21 q = + 2 + 15 + 150 + 1707 + 20,910 (4.28) + 268,616 + 3,567,400 + ....

Define the theta functions

OO ths(q)-- E (-1)J q(Sj't'l)2/12' (4.29) and

B 2 - AI/3 [ths(q)] 2, (4.30) where A comes from (4.15) and (4.16). Then the roots of the 3acobi sextic (4.15) are

5 [ths(qS)] 2 Svo -- B2 , (4.31a)

[ths(el2kql/S)]2 (0 < k < 4), (4.31b) Sk -- B2 , where (4.32) Icosahedral symmetry 27

The roots of the Jacobi sextic obtained from equations (4.31) should be checked before proceeding further, in order to assure that the correct square root of k ', the correct cube root of A and the correct fifth root of q have been taken for Equations (4.27), (4.30) and (4.31b), respectively. After the correct roots soo and st (0 < k < 4) for the Jacobi sextic (4.15) have been obtained, these are used to calculate the roots of the Brioschi quintic (4.10), by using the two equations

1 s y~ = --~ ( oo - st) (st+2 - st+s)(st+4 - st+l), (4.33)

Z 2 Yt = (y~)2 + (10/A) y~ + (45/A2) ' (4.34) adding indices modulo 5 in Equation (4.33) and using (4.34) in order to assure that the correct square root is taken in Equation (2.20). Undoing the Tschirnhaus transformations gives

+ P Yt (4.35) = - 3' from (4.9), for the roots of the principal quintic (4.3) and then

E+ (z:~ - v)(u 3 +Au 2 + Bu + C) + (zt - v) 2 (2u+A) (4.36) x~= u4+AuS+Bu2+Cu+D+(z t_v)(3u2+2Au+B)+(zt_v)z , for the roots of the general quintic (4.1). The above algorithm has been checked by implementing it as a complete Pascal program which executes on a personal microcomputer. This computer program also checks the ambiguities arising from the several radicals in the algorithm, by investigating whether the product

4 (s - soo) ~-~(s - st), (4.37) k--O agrees with the original Jacobi sextic (Equation (4.15)) for all of the possible choices of V~, ql/~ and A I/3 in (4.27), (4.31b) and (4.30), respectively, before proceeding with the final steps of the algorithm involving the sequence of roots st ~ Yt .~ zt --* zt using (4.33)-(4.36) and the correct values of st.

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